Convergence of Sequences and Series - My Brain just doesnt get it

Hello folks. First off i want to say this is an awesome forum! But lets get to the topic:

I totally understand the concept of convergence/divergence and i dont have problems to follow the lessons. But....
when it comes down to exercises/exams i totally fail in showing wether a series/sequence converges or not

for example:

i got the series:

my thinking was... finding a series a_n greater than the given one and then proof its convergence. but i seem to fail ....

another typical problem would be:

testing for convergence as n goes to infinity

my thought was writing the 1 as and then have raised to the power of 4n

well a huge problem might be that the exercise says: "Use the comparison test" (Majoranten/minorantenkriterium in german)

Are you sure it is not for instance? Or did you consider similar series right before, like , that could be re-used? (This wouldn't help performing a comparison : here, this is impossible. But other tricks may come in handy)